PERFEZIONAMENTO IN MATEMATICA FINANZIARIA E ATTUARIALE. The impact of contagion on large portfolios. Modeling aspects.

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1 SCUOLA ORMALE SUPERIORE PISA PERFEZIOAMETO I MATEMATICA FIAZIARIA E ATTUARIALE TESI DI PERFEZIOAMETO The impact of contagion on large portfolios. Modeling aspects. Perfezionando: Marco Tolotti Relatori: Ch.mo Prof. Paolo Dai Pra Ch.mo Prof. Wolfgang J. Runggaldier

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3 A Maria e Federico

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5 Acknowledgments I must rst express my deepest gratitude towards Paolo Dai Pra and Wolfgang Runggaldier for their precious help and their courtesy in reading countless versions of this thesis. Their suggestions and guidance have always encouraged me throughout the study. A very special thanks goes out to Elena Sartori. I cannot forget the beautiful hours spent discussing about the generators of the Markov processes. I am very grateful for the valuable comments of Rama Cont and Rüdiger Frey. I wish to thank Maurizio Pratelli for his patience and kindness, the Amici della Scuola ormale and the Scuola ormale Superiore for the nancial support. I was delighted to interact with the Finance group at the Department of Mathematics of the ETH Zurich and the Swiss Banking Institute of the University of Zurich during the Master of Advanced Studies in Finance. I would like to thank in particular Philipp Schönbucher for giving me the opportunity to work with him for the Master thesis. My gratitude goes also to Fulvio Ortu for his trust in oering me a position at the Bocconi University. I must acknowledge the (former) Istituto di Metodi Quantitativi and the Department of Finance at the Bocconi university for the nancial support. Anna Battauz, Elena Catanese, Francesca Beccacece, Giuliana Bordigoni, Marzia De Donno, Claudio Tebaldi, Fabio Maccheroni, Francesco Corielli, Gianluca Fusai and Gino Favero have been supporting me with inspiring discussions and their friendship.... no words to express how I am grateful to Maria and my parents. Milano, June 28

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7 Contents Introduction 9 Credit quality and credit risk 5. Default probability and loss given default Dierent approaches to credit risk modeling Credit migration Modeling dependence Portfolio credit risk Basic denitions and concepts Basel II insights Conditionally independent models Mixture models Threshold models A view on dependent credit migration models Contagion models Large portfolio losses Conditionally independent static models A static model with local interaction A conditionally Markov dynamic model Dynamic models with random environment Exogenous (static) random environment Some general results on large deviations The model for contagion Implementation of a Large Deviation Principle A law of large numbers for portfolio losses Simulation results

8 8 3.5 A central limit theorem Endogenous (dynamic) random environment The model in details Invariant measures and non-reversibility Studying the dynamics of the system Deterministic limit: Law of large numbers Equilibria of the limiting dynamics: Phase transition Fluctuations: A central limit theorem Convergence of generators approach A functional approach Applications to portfolio losses Simulation results Conclusions 5 A Technical proofs (Chapters 3-4) 7 A. Proof of Theorem A.2 Proof of Theorem A.3 Proof of Theorem A.4 Proof of Proposition A.5 Proof of Proposition A.6 The eigenvalues of the matrix A Bibliography 3

9 Introduction This Ph.D. thesis is part of a research project originated within the Probability group of the Mathematics department of the University of Padova in the year 25. This project involves the Professors Paolo Dai Pra and Wolfgang J. Runggaldier, a Ph.D. student of the University of Padova, Elena Sartori and Marco Tolotti of the Bocconi University and the Scuola ormale Superiore. The goal of this project is to integrate techniques and skills coming from dierent disciplines (Probability, Statistical Mechanics, Finance, Econometrics, Physics) in order to describe, analyze and quantify phenomena related to current issues in the Finance literature. Financial motivations The crucial issue that has motivated in particular this thesis is to describe a mathematical framework within which it is possible to explain the nancial phenomenon referred to clustering of defaults or credit crisis. By clustering of default we mean the situation in which many obligors experience nancial distress (default or downgrading in a rating system) in a short time period. What we mean by many defaults in a short time will be discussed later. It is clear that in order to speak of a credit crisis there must be an unexpected breakdown from the standard economic business cycle. In some sense we have in mind a sudden change in the equilibrium of the credit market. Financial distress, default, downgrading are all issues belonging to the eld of credit risk management. Managing the risk concerns the identication and the analysis of the randomness intrinsic in the nancial world and in particular the capacity to predict and quantify the losses triggered by changes in the variables describing the nancial market. More specically, when speaking of credit risk, one is dealing with the risks connected with the possible changes in the credit worthiness of the obligors. We shall concentrate, in particular, on the losses related to large portfolios, meaning to portfolios of many obligors with similar characteristics. The precise meaning of words such as large and similar shall be extensively discussed. When dealing with many obligors, the issue of modeling the dependence structure plays a major role. Our idea is that a credit crisis could be explained as the eect of a contagion process. A rm experiencing nancial distress may aect the credit quality of business 9

10 ITRODUCTIO partners (via direct contagion) as well as of rms in the same sector (due to an information eect). Therefore the mechanism of credit contagion is the crucial mechanism that we are going to develop in order to describe clustering of defaults. Reduced form models for direct contagion can be found -among others- in Jarrow and Yu (2) [47] for counterparty risk, Davis and Lo (2) [22] for infectious default, Kiyotaki and Moore (987) [48], where a model of credit chain obligations leading to default cascade is considered, Horst (26) [46] for domino eects, and Giesecke and Weber (25) [4] for a particle system approach. Concerning the banking sector, a micro-economic liquidity equilibrium is analyzed by Allen and Gale (2) []. Recent papers on information driven default models are e.g. Schönbucher (23) [59], Due et al. (26) [28], Collin-Dufresne et al. (23) [2]. An important point that we would like to stress is that we are aiming at describing the formation of a credit crisis starting from the inuences that the single obligors have among each others, in other words as a microeconomic phenomenon. The standard literature suggests that the aggregate nancial health of the system is fully described by some macroeconomic factors that capture the business cycle and then inuence the credit quality of the obligors. These factors are usually exogenously specied. One consequence of our micro" point of view is that in our modeling framework a global health indicator is endogenously computed and not a priori assigned. otice that the same philosophy is already standard practice in other disciplines: similar models and techniques as the ones developed in this thesis are applied for instance by Brock and Durlauf in (2) [8] for modeling social interactions or by Cont and Löwe (998) [5] in order to describe phenomena as herding behavior or peer pressure. The last remark is about a new trend in the literature concerning portfolio credit risk. In the last years portfolio credit derivatives such as default basket and Collateralized Debt Obligations (CDO's) have become very popular. In order to treat this kind of structured derivatives a dynamic study of the aggregate losses L(t) caused by the defaults in the underlying pool of assets becomes crucial. It has been documented that an eective study of the dynamics of the aggregate losses L(t) may not necessarily require a full understanding of the single name processes related to the underlying assets. For this reason a new approach has been recently proposed in the literature: the so called top-down approach (for more details see Cont and Minca (28) [6], Giesecke and Goldberg (27) [39] and Schönbucher (26) [6]). When considering Markov chain models similar in spirit to the ones proposed in this work, it is possible -under certain hypothesis- to fully explain the evolution of the system via aggregate sucient statistics. As argued also in Frey and Backhaus (27) [36], this is exactly the philosophy behind a top-down model. In particular we shall see in Chapter 3 (see Remark 3.5.8) that our approach may be considered as a useful tool also under this new perspective since it naturally exploits the problem of computing approximations of L(t) in a parsimonious way as a function of aggregate asymptotic variables that may account for the heterogeneity in the underlying portfolio. We have thus identied the nancial core of this discussion: quantifying the losses connected to the deterioration of credit quality, taking the contagion into account and eventually describing under which market conditions a credit crisis may take place.

11 Technical aspects From a more technical point of view there are dierent tasks that we have to deal with; we briey illustrate the most important ones: Having large portfolios in mind, we are going to describe asymptotic results for a system of innitely many rms and then provide nite volume approximations. It basically implies the development of suitable Laws of Large umbers (LL) and Central Limit Theorems (CLT). The implementation of LL and CLT for the study of large portfolios is not new in the risk management literature; for example it is implemented (in a rather basic setup, without considering contagion) in the Basel II accord. More recent generalizations to contagion models have been proposed -among othersby Frey and Backhaus (26) [36] and by Giesecke and Weber (25) [4]. A dierent approach to the study of large portfolio losses may concern the extreme events (analyzing the tails of the loss distributions). In this case, it is quite common to rely on Large Deviation techniques. For a recent survey on these techniques, applied to Finance and risk management, see Pham (27) [56]. In Dembo et al. (26) [23] an application to large portfolio losses is proposed. The issue of contagion may be described relying on interacting particle systems borrowed from Statistical Mechanics. otice that also in the literature of quantitative risk management the terminology interacting intensities is used in order to describe reduced form models where interaction is taken into account. The use of particle systems is rather common in the Social Sciences literature, for instance when modeling social interactions and aggregate behaviors (see the paper by Cont and Bouchaud (2) [5] for a discussion on herding behavior in Finance). Particle and dynamical systems can be found also in the literature on nancial market modeling. It has been shown that some of these models have "thermodynamic limits" that exhibit similar features compared to the limiting distributions (in particular when looking at the tails) of market returns time series. For a discussion on nancial market modeling see the survey by Cont (999) [4] and the papers by Föllmer (994) [33] and Föllmer et al. (24) [34] that contain inspiring discussions on interacting agents. In order to derive a LL and a CLT for a particle system we rely on Large Deviation techniques. The idea is to consider M the space of probability measures on trajectories endowed with the Skorohod topology. It is quite easy to state a Large Deviation Principle (LDP) for a reference system where there is no interaction between the rms. Then the goal is to nd a suitable function F : M R + that relates (via Varadhan's lemma) the LDP for the reference system with our interactive model for contagion. This technique has been applied to spin-ip systems by Dai Pra and Den Hollander (996) [8]. Concerning the central limit theorem we shall develop two dierent approaches. The rst one (based on large deviations) relies on a theory by Bolthausen (986) [4]. The second one is based on a weak convergence-type approach based on the uniform convergence of the generators of the associated Markov chains (as developed in the book by Ethier and Kurtz (996) [32]).

12 2 ITRODUCTIO We are aiming at building a dynamic model. This has to be done in order to describe the time evolution of the variables describing the credit quality of the obligors, and hence (under particular conditions) the formation of a credit crisis. We shall propose a model where the credit crisis is connected with the existence of multiple equilibria for the dynamical system. The system may in fact spend some time near an unstable conguration and then suddenly decay to a stable one. We shall see that this eect is related to the phase transition, i.e., on the level of interaction in the model. It is nally worth to spend some words on the point of view that we have adopted in this research project (at least in its rst step). We have focused our attention on the modeling aspects, trying to build a model as simple as possible where the eects of contagion (as the credit crises) could have been observed. Moreover we have looked for a completely solvable model, where closed form solutions can be provided. Put dierently, we have given more relevance to the technical part of the problem and the modeling aspects. On the other hand we have not developed the validation part of the model, i.e., the calibration and the analysis of real data. evertheless many numerical simulations are provided in order to illustrate the shape of the loss distributions and the trajectories of the health indicators under dierent market conditions. Although this observation could be seen as a drawback of our work, we would like to argue that what we are aiming at are the qualitative aspects more than the quantitative ones (at least in the st step of the research). Indeed, the models we are going to propose are very basic and have to be considered as a starting point for the construction of more realistic models that may better t real data. Structure of the thesis and main results The thesis is divided into ve chapters and one appendix. Chapter and Chapter 2 are devoted to the introduction of the main concepts and the basic tools for managing credit risk. In writing this introductive part we had two goals in mind: rstly we aimed at letting this work be -as much as possible- self consistent. On the other hand we have tried to report and briey discuss the main approaches and models used in the literature. We have focused in particular on those models that can be considered as the starting point for our research. The guideline to these two chapters may be summarized in the three following fundamental questions: How can we model the credit quality of one obligor? How can we determine her default probability and the possible losses that the event of default may trigger? Given a set of dierent obligors, how can dependence be modeled and eventually joint default probabilities computed? How can we model credit contagion? Is it possible to build a model that explains clustering of defaults (credit crises)?

13 3 These questions shall be picked up again in more detail in the two introductive chapters and should nd appropriate answers in the further exposition. otice that they go from the very basic concepts up to the issues that have motivated our research. Hence these questions (and the two chapters themselves) are intended to build a bridge between the existing literature and our point of view. The succeeding two chapters contain the mathematical implementation and the discussion of our two original models, developed in order to tackle the problem of contagion in a credit risk perspective. In Chapter 3 we propose in particular a rst attempt to model an interactive system of defaultable counterparties where a local random environment enters into play. We shall see that a crucial object of this work is the so called empirical measure. Indeed, suppose that rms are acting in a market and their default indicators σ(t) = (σ i (t); i =,..., ), where σ i (t) {, }, evolve in time. We denote by σ[, T ] D[, T ] a trajectory on the interval [, T ] of such a rating indicator. D[, T ] denotes the space of càdlàg functions endowed with the Skorohood topology. The empirical measure ρ is dened as follows ρ (σ[, T ]) = δ (σi [,T ]). It is a random measure taking values in M (D[, T ]), the space of probabilities on D[, T ]. ρ weights the realizations of the dimensional process σ[, T ]. Put differently the empirical measure can be thought of the physical (historical) measure of the market. Most of the mathematical results of this thesis are concerned with the sequence of measures (ρ ). In this chapter we state in particular a large deviation principle for this sequence of measures (Theorem 3.3.3) and then we prove a suitable law of large numbers (Theorem 3.3.6), nding a unique Q such that ρ Q almost surely. Signicant and very useful for applications is also the characterization of Q provided in Proposition In Section 3.5 we nally state and prove a functional central limit theorem characterizing the uctuations of ρ around Q (Theorem 3.5.6). The proof of this theorem is inspired by the seminal work of E. Bolthausen (see [4]). In [4] a rather general framework is proposed in order to derive central limit theorems in Banach spaces. Since M is not a Banach space we are forced to construct an auxiliary space where a suitable large deviation principle is inherited and a central limit theorem can be proved accordingly. This procedure is summarized in Theorems and Although this rst model shows some interesting features and -to some extentgeneralizes the present literature on dynamic mean-eld models for large portfolio losses, it turns out to be not yet comprehensive enough in order to explain clustering and credit crises.

14 4 ITRODUCTIO Chapter 4 is then devoted to the analysis of a new framework that makes possible the explicit identication of the desired clustering eect. To this aim we introduce a fundamental indicator of robustness ω i {, } that is coupled with σ i dened before. The 2 state variables (σ, ω) evolve in time and their dynamics turn out to be non trivial at all. In particular our model is non-reversible. As compared to similar but reversible stochastic interacting systems, more careful arguments have to be used in order to prove the large deviation principle, which represents the basic tool in our approach. Similarly to Chapter 3, the rst main result is a Law of Large umbers (Theorem 4.3.2) based on a Large Deviation Principle (Proposition 4.3.4). In Theorem 4.3. we shall see that dierent asymptotic congurations can be found, depending on the values of the parameters. This phenomenon (called phase transition) has implications for the description of a credit crisis as we shall explain in more details in Chapter 5. The last section of this chapter is devoted to the study of the uctuations of the empirical measure around its limit. Two dierent approaches are described, the former is based on uniform convergence of generators (Theorem 4.4.). The latter (Theorem 4.4.5) mimics the functional approach already introduced in the previous chapter. One remark is needed at this point. Part of these rather technical proofs have been pursued in collaboration with Paolo Dai Pra and Elena Sartori. We shall refer to the Ph.D. thesis of Elena where some explicit computations have not been reported in this dissertation. To make the exposition less heavy we have postponed to Appendix A the most technical proofs of Chapters 3 and 4. The nancial applications are discussed in Chapter 5. The main result of this chapter is Theorem It is concerned with the computation of risk measures for managing the risk involved in large portfolios. Various examples are provided, some of them have been suggested by the existing literature on the subject. The second issue that we are going to analyze in this chapter is related to the formation of a credit crisis. To this aim the equilibria of the limiting dynamics and their stability are studied and the phase transition is fully characterized. At the end of the chapter, dierent graphs and numerical implementations are presented in order to support the nancial interpretation of our model. Finally we conclude this thesis with a brief summary of the main results and mentioning some open problems and possible lines of future research in this area.

15 Chapter Credit quality and credit risk The goal of the rst two chapters of this thesis is to equip the reader with the basic notations and concepts necessary to enter the world of credit risk. First of all we would like to specify what we intend for "risk" in the context of credit and propose a mathematical framework in which a punctual quantitative analysis can be pursued. A rather concise denition of risk in nance could be as follows: we speak about risk when we consider the possibility of having unexpected changes in the variables that describe the nancial model. We thus have to dene a suitable probability space {Ω, F, P } that summarizes the states of the world, the interesting events and a possible probability measure on them. We may possibly add a ltration (F t ) t that describes the ow of information when considering dynamic models. Finally we dene random variables (eventually processes) X : Ω R, representing the nancial variables that we are interested in. In the context of credit risk, X should basically describe the credit quality of a given obligor, where an obligor is somebody who has to pay back a debt to somebody else in the future. For credit quality we mean the ability of being able to pay back obligations. How can we model the credit quality of one obligor? How can we determine her default probability and the possible losses that the event of default may trigger? Suppose that this obligor is a rm; in this case X could represents the value of the rm at a given time. Thus we could be interested in estimating the probability that X falls below a certain value (also named a threshold level) and this could be a good choice for a credit quality indicator. In an even naiver world, X could simply be an indicator of default (bankruptcy): if X = the rm is able to pay back its obligation, if X = it is not. All these issues will be developed and specied in the following chapters. In particular many dierent models with dierent specications for X will be provided as well as dierent specications for {Ω, F, P } and (F t ) t. We would like to stress the fact that the majority of the concepts we are going to state in the rst two chapters are not new and can be found in dierent books dealing with credit risk. Our purpose is to give a glance on the main aspects, focusing the attention on the basic building blocks necessary to assess and quantify the riskiness of a portfolio of obligors and to price credit derivatives. We shall describe the existing techniques used to compute these building blocks and the relevance of particular 5

16 6 CHAPTER. CREDIT QUALITY AD CREDIT RISK assumptions often used to provide explicit formulae. In doing this we shall build the bridge between existing literature and our modeling ideas, leading the reader to the new framework that we have introduced in order to solve some open problems that are still debated in the present literature.. Default probability and loss given default In this section we briey recall the very basic tools used to deal with credit risk. Our aim is to focus the attention on the so called building blocks, i.e., the basic objects on which the risk measurement and the pricing techniques of large part of more complicated securities rely on. These objects are the so called default probability and the loss given default. In a credit risk environment the basic concept is default. We do not enter into the discussion of the bankruptcy procedures. We simply say that a default time τ has to be dened. In the next section we shall discuss on how τ is characterized within dierent models. Our concern now is simply that τ is a random time and in particular a stopping time with respect to its own ltration (H t ) t. Default probability (PD) We dene the default probability between t and T for the obligor as p(t, T ) = P (τ T τ > t). (.) It is worth to dene also the conditional version for the default probabilities meaning the probability at time t of having a default between future dates T and T 2 knowing survival up to T : p(t, T, T 2 ) = p(t, T 2) p(t, T ), t T T 2. These quantities are then used in order to determine an important object often referred to the hazard rate. We dene the hazard rate between t and T as h(t, T ) = lim t p(t, T, T + t) t whenever this limit exists. Remark.. It should be stressed that in more general models (for example if the interest rate r is stochastic or if other random variables inuence the market prices) one has to be careful with the denition of hazard rates. In particular it is very important to study measurability conditions (namely which ltration or information structure are available in the model) and how the total information is related to the ltration generated by the default process (see chapters 5,6 in [3] for a punctual discussion). In particular it can be shown (see for instance Section 6.2 in [3]) that for a random time τ one may consider two dierent hazard processes (both well dened and mathematically meaningful) and they may not coincide if some regularity hypotheses are not made on the model. How F and H are related is a crucial point. An extensive discussion is made in [3]. We do not enter into details at this point.

17 .2. DIFFERET APPROACHES TO CREDIT RISK MODELIG 7 Formally we can also dene a local hazard rate: γ(t) := h(t, t) = lim p(τ t + t τ > t). t t Loss given default (LGD) Suppose that a default happens. Suppose moreover that a bank holds a position issued by the defaulting obligor. At the time of default τ < T part of the investment of the bank is lost due to the impossibility of the obligor of paying back obligations. The part that the bank (or more general an investor) cannot recover is called loss given default. This quantity is often modeled as a random variable l = δe where δ represents a random proportion of the exposure that is lost and e the actual exposure at default. A rather general discussion on pricing issues of defaultable claims is also proposed in [3]. There, it is shown that (PD) and (LGD) are the building blocks, necessary to price many types of derivatives such as defaultable xed-coupon bonds, credit default swaps (and many variants of them), asset swap packages..2 Dierent approaches to credit risk modeling Up to now we have considered a given probability space {Ω, F, (F t ) t, P } endowed with a market ltration. On this ltered probability space we have dened a default time τ without taking care of the economic process leading to this denition and without specifying any distributional assumption on it (except for some assumptions on the corresponding ltration). We now have to implement models that may help in computing the building blocks seen in the previous section. In the literature two typical and rather dierent point of views are developed. The rst class of models, the so called structural models (or rm value models), relies on the precise denition of some economical variables (such as the asset value process for a rm). The evolution of these variables determine the credit quality of the obligor itself. The second class of models, the reduced form models (or intensity based models), is based on a more statistical point of view. The idea here is that -having dened a suitable family of processes (without direct economical interpretation)- one tries to calibrate the parameters in order to t historical time series or other data. These two approaches lead of course to dierent modeling frameworks. Both have important advantages: the former gives more direct intuition on what is going on economically; the latter is usually easier to implement and allows for more freedom in modeling parameters and in tting data 2. 2 We would like to stress the fact that the dilemma" on what the best philosophy" should be, is far from being solved. Moreover, notice that it involves the fundamental debate whether when using random variables one is or is not allowed to forget" about the underlying probability space (the underlying experiment") taking into account only distributional consequences (the numerical results of the experiment).

18 8 CHAPTER. CREDIT QUALITY AD CREDIT RISK A. Structural models The basic idea behind structural models is to consider the evolution in time of an underlying process X that is related to some fundamental indicators of the rm. Then, usually, the default happens when this process hits a predetermined (possibly stochastic) barrier D. So that τ is dened as a rst passage stopping time, indeed τ = inf t {X t D}. (.2) In the progenitor of these models, due to Merton (974) [53], the underlying process was the rm asset value process V t. In particular it is assumed that V t = S t + B t where S t is the value at time t of the equity and B t is the value at time t of a zero coupon bond with face value B and maturity T. In this basic context where the payments to the bond-holders are due at a xed time T, the event of default is even easier to describe, than in Equation (.2). We have in fact that default happens if and only if V T < D so that τ = T. Moreover the recovery at default is simply B T = min(v T, B). Suppose that V t evolves according to the dierential equation dv t = µ V V t dt + σ V V t dw t where V = V >, µ V R and σ V > are constants and W is a standard Brownian motion. Then one can compute the default probability ( ln(b/v ) (µv σv 2 p(, T ) = P (V T < B) = /2) ) σ V T where ( ) stands for the standard normal distribution function. We do not want to enter into more details on this topic, we simply mention the fact that this modeling idea is still used in practice. More sophisticated models have been proposed, we shall see some extensions in the next sections (the KMV model, credit migration models, a multivariate extension of Merton model,...). What remains a milestone in all of them is the precise reference to an underlying explanatory (fundamental) process and the fact that the default time is an hitting time 3. B. Reduced form models We have seen what a model for credit risk should be able to provide: probabilities of default and losses given default. In the structural approach, we have seen a basic methodology to compute them. One dierent approach is to consider models where the family of the distributions for the probabilities of default is a priori given and the model parameters are then 3 The fact that τ is an hitting time for a continuous path process makes the time τ be fully predictable. In particular this implies that the local probability of default h(t) is always zero. We shall see that this is not the case in the so called intensity based models (and this reects better real data). This is probably the main (mathematical) dierence between the two approaches and the reason that makes intensity based models so popular. Many authors have tried to relax this hypothesis in order to make τ totally inaccessible. We refer to [58] for a discussion on the accessibility of a stopping time and to [22], [], [9], [43] and [44] for dierent bridge" models between structural and reduced form models.

19 .3. CREDIT MIGRATIO 9 computed in order to t market data. In particular it is useful to consider models where a local probability of default is well dened. As in the previous sections we assume that a probability space {Ω, F, P } is given. We consider the nonnegative random variable τ : Ω R +, with P {τ = } = and P {τ > t} > for all t, that describes the default time. For simplicity we suppose that the information available to the market is the natural ltration generated by the default time τ. Indeed, we consider the σ-elds H t := σ({τ u} : u t) and the corresponding ltration H = (H t ) t. In many reduced form models the default time τ is distributed according to an exponential law, so that the event of default is related to the rst jump of a Poisson process ((t)) t such that τ = inf{t : (t) = }. After the rst jump we stop the Poisson process so that for t > τ we have (t) = (τ) =. Summarizing we nally have (t) = (t τ) = I {τ t}. otice that H t = σ((u) : u t). We denote by F (t) = P {τ t} for all t R + the cumulative distribution function of the default time. Assumption.2. F (t) is absolutely continuous with respect to the Lebesgue measure, that is, it admits a density function f(t) : R + R + such that F (t) = t f(u)du, then F (t) = e t γ(u)du, where γ(t) = f(t)( F (t)). The function γ is called intensity function. It can be shown that in this case γ(t) := lim P (τ t + t τ > t). t t Put dierently, it describes the local probability of default. In the class of models where the intensity function is well dened, it is natural to assume γ as the primitive object to be characterized. These models are usually called intensity based models..3 Credit migration In the previous sections we have described the basic tools when dealing with default risk. In particular we have dened a stopping time τ as the time of default. We want now to consider more general frameworks in which the default event is no longer the only interesting event (and the unique risk to be taken into account). Instead of looking only at the default event, we shall describe the credit quality of an obligor considering a whole set of rating classes where the default is only the last (and worst) class. We are thinking of the so called credit rating models. These models are implemented by rating agencies (Moody's, Standard and Poor's, Fitch) but also by the internal rating systems of nancial insitutions 4. The basic idea behind these models is to assign each obligor to one class that is characterized by an estimated probability of default (i.e. of falling down to the default state). This means that all the available information about the probability of default of each obligor is given by its rating class. 4 In the new Basel II accord the single institutions are encouraged to implement internal systems to asses credit worthiness of obligors, see [2].

20 2 CHAPTER. CREDIT QUALITY AD CREDIT RISK For example in Moody's model the rating classes are labelled from the best to the worst as AAA, AA, A, BBB,..., CCC, D, where D indicates the default state. Thus we have to estimate a K-dimensional vector of probabilities of default where K is the number of rating classes (apart form default). otice that all the migration probabilities have to be determined, in the sense that we are interested in the probability that a rm in class AA falls within a year into class B, and this has to be done for each pair of classes. This means that we have to determine a transition matrix in which the entries are exactly the probabilities of migration among the K + classes. To formalize this issue we give the following denitions: Denition.3. We label the K + classes as K, K,...,, where k = indicates the default state; We name K the set of all the rating classes and K = K /{} the set of all the classes except for default; Dene the rating process R {t } as the process R(ω, t) : Ω [, ] K where {Ω, F, F t, P } is a probability space where F t is the market ltration; We dene Q(t, T ) R K+ R K+ as the transition probability matrix for the time interval [t, T ], so that (Q(t, T )) ij = P (R(T ) = j F t R(t) = i). otice that (Q(t, T )) ij, i, j K ;, j K and Q(t, t) = I where I is the identity matrix. K j= Q(t, T ) i,j =, i K ; Q(t, T ),j = The problem of credit migration is now to specify a model in order to describe the evolution of the transition matrix Q(t, T ) and eventually to calibrate it to the data. In the standard literature as well as in the most famous models used in practice, the rating transition process R is assumed to evolve as a time homogeneous Markov chain. Basically this means that (Q(t, T )) ij = P (R(T ) = j R(t) = i); Q(t, T ) = Q(T t) t T. where Q is a transition probability that depends only on the length of the time interval between t and T. The Markov hypothesis simplies the computation of the migration probabilities. It can be proved (see [58] for more details) that in this case Q(t, T ) = exp{(t t)λ} where the exponential is dened formally as the limit of the series expansion exp{(t t)λ} = ((T t)λ) n n= n!